\documentclass[11
pt]{amsart}
%\usepackage{amssymb,amscd}\textheight=23cm


\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{epstopdf}
\usepackage{amsthm}
\usepackage{times}

\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   % ... or a4paper or a5paper or ... 
%\geometry{landscape}                % Activate for for rotated page geometry
%\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
%\usepackage{amssymb}
%\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth=15cm
%\titlepage
\topmargin=0pt
\headheight=0pt
\headsep=0pt
%\footskip=0pt
%\oddsidemargin=0pt
%\marginparsep=0pt
%\marginparwidth=0pt
\pagestyle{empty}

\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\N}{{\mathbb N}}

\newcommand{\T}{{\mathcal T}}
\newcommand{\B}{{\mathcal B}}
\newcommand{\X}{{\mathcal X}}

\newcommand{\vx}{{\mathbf x}}
\newcommand{\vy}{{\mathbf y}}
\newcommand{\vz}{{\mathbf z}}

\begin{document}
\parindent=0pt


{{\noindent Math 5510} \hfill {Homework 7-- {\bf  Due  Monday  12/4/17}}
\hfill {\ \ \ \ }\hfill}



\vskip.2cm

 \hrule



\bigskip

\begin{enumerate}

\item (2 pts)     \emph{Geodesic curvature of parallels in $S^2$}: As usual $S^2$ is the unit sphere in $\R^3$.  It can be parametrized by spherical coordinates $\phi, \theta$, where $0\le\phi\le\pi$ is co-latitude and $0\le \theta\le 2\pi$ is longitude:
\begin{equation}
\label{eq:spherical}
(x,y,z) = (\sin\phi \cos\theta, \sin\phi \sin\theta, \cos\phi)
\end{equation}
%\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   %\centering
    %\includegraphics[width=3in]{Sphere.pdf}
  % \caption{}
  % \label{fig-threedimcomp}
%\end{figure}
The parallels are the curves $\phi = c$, $0\le c \le \pi$ a constant.  The curves $\phi =0, \pi$ degenerate to the north and south pole respectively, and $\phi = \pi/2$ is the equator.  \emph{Compute the geodesic curvature $|\frac{D\gamma'}{Ds}| = |\gamma''^T|$ of the parallel $\gamma = \gamma_\phi$ for each value of $\phi$}

\emph{Suggestion}:
\begin{enumerate}
  \item For fixed $\phi$, (\ref{eq:spherical}) gives a curve with parameter $\theta$, which is not arc-length unless $\phi  = \pi/2$, the equator.  Re-parametrize this curve by arc-length $s$, get a curve $\gamma(s)$ (depending on $\phi$)
  \item Figure our the tangential component $\gamma''^T$ and find its magnitude (answer: $\cot\phi$)
  
\end{enumerate}

\item (4 pts) \emph{Gaussian curvature of the sphere of radius $R$}: Let $S^2(R)$ denote the sphere in $\R^3$ of radius $R$ and let $\vx(\phi,\theta)$ be the parametrization of $S^2(R)$ obtained by multiplying (\ref{eq:spherical}) by $R$.  Thus $\vx:[0,\pi]\times [0,2\pi]\to S^2(R)$ is surjective but maps $0\times [0,2\pi]$ to the north pole, etc.
\begin{enumerate}
  \item Not to forget topology, prove that $\vx$ is an identification.
    \item Write the expression for $ds^2 = d\vx\cdot d\vx$ for this parametrization.
  \item Let $r$ be distance from the north pole.  Express $\phi$ as a function of $r$ and change the expression for $ds^2$ to one that is a function of $r$ and $\theta$.  It should be of the form $dr^2 + g(r)^2 d\theta^2$ for a suitable function $g(r)$.   Observe that the circumference of a circle centered at the north pole of radius $r$ is $\int_0^{2\pi}g( r)  d\theta = 2\pi g(r)$.
  \item Show that the Taylor expansion of $g$ at $r=0$ begins
  $$
  g(r) = r - \frac{r^3}{6R^2} + \dots
  $$
The quantity $1/R^2$ is  called the \lq\lq Gaussian curvature" of $S^2(R)$.  It measures the deviation of the circumference of circles of radius $r$ from the value $2\pi r$ in Euclidean geometry.
\end{enumerate}

\item (4 pts) \emph{Gaussian curvature of a cylinder}:  Let $C$ be the cylinder $\{x^2 + y^2 =1\}\subset \R^3$. \ 
 \lq\lq Parametrize " $C$ by the map $\vx: \R^2\to C$ defined by
$$
\vx(u,v) = ( \cos u, \sin u, v).
$$
\begin{enumerate}
  \item Even though $\vx$ is not injective, show that $\vx$ is locally injective, is an open map, and is an identification.  So it can be used to compute $ds^2 = d\vx\cdot d\vx$.

  \item Find $ds^2 = d\vx\cdot d\vx$.  Any surprises?
\item Find the circumference of the geodesic circles of radius $r$, for small enough $r$,  so that these curves will not intersect themselves. What's the Gaussian curvature?
\item Find all geodesics in $C$ joining $(1,0,0)$ to $(1,0,2\pi)$.  Show that there are infinitely many distinct ones.  Draw a picture.
\end{enumerate}
 
 \item \emph{Extra credit problems on the torus}
 
 Let $T$ be the torus $\R^2/(2\pi \Z)^2 = \R^2/\sim$, where $(x,y)\sim (x',y')$ if and  only if $x-x'$ and $y-y'\in 2\pi\Z$.  Can measure lengths of curves in two ways:
 \begin{itemize}
  \item $ds_1^2 = dx^2 + dy^2$ on $\R^2$, is invariant under translations, is well defined on $T$.
  \item $ds_2^2 = d \vx\cdot d\vx$ where $\vx(\phi,\theta) = ((2 + \cos\phi)\cos\theta,(2+\cos\phi)\sin\theta,\sin\phi)$. Then  $\vx:\R^2\to \R^3$ and $\vx(\phi+2 \pi m,\theta +2\pi n) = \vx(\phi,\theta)$ for all $m,n\in\Z$, therefore $\vx$ gives a map $\vx:T\to\R^3$ with image the familiar torus of revolution in $\R^3$/
\end{itemize}
%\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
  % \centering
    %\includegraphics[width=2.5in]{TorusR3.pdf}
  % \caption{}
  % \label{fig-threedimcomp}
%\end{figure}

\begin{enumerate}
  \item For $(T,ds_1^2)$ find the Gaussian curvature and all the geodesics from $[(0,0)]$ to $[(\frac{1}{2},\frac{1}{2})]$.
  \item For $(T,ds_2^2)$ show that all circles $\theta = const$ are geodesics, and decide which of the circles $\phi = const$ are geodesics.
  \item Prove that there is no isometry between $(T,ds_1^2)$ and $(T,ds_2^2)$.
\end{enumerate}
 \end{enumerate}  
   
 

                

 \end{document}
